P. 161 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

umgr2edg1 16001*

If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 8-Jun-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥))

Theorem

usgr2edg1 16002*

If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 8-Jun-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥))

Theorem

umgrvad2edg 16003*

If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 16000. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)

⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐸 (𝑥 ≠ 𝑦 ∧ 𝑁 ∈ 𝑥 ∧ 𝑁 ∈ 𝑦))

Theorem

umgr2edgneu 16004*

If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex, analogous to usgr2edg1 16002. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)

⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ 𝐸 𝑁 ∈ 𝑥)

Theorem

usgrsizedgen 16005

In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 7-Jun-2021.)

⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺) ≈ (Edg‘𝐺))

Theorem

usgredg3 16006*

The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ (𝐸‘𝑋) = {𝑥, 𝑦}))

Theorem

usgredg4 16007*

For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Theorem

usgredgreu 16008*

For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Theorem

usgredg2vtx 16009*

For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.)

⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theorem

uspgredg2vtxeu 16010*

For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.)

⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theorem

usgredg2vtxeu 16011*

For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)

⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theorem

uspgredg2vlem 16012*

Lemma for uspgredg2v 16013. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉)

Theorem

uspgredg2v 16013*

In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Theorem

usgredg2vlem1 16014*

Lemma 1 for usgredg2v 16016. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)

Theorem

usgredg2vlem2 16015*

Lemma 2 for usgredg2v 16016. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))

Theorem

usgredg2v 16016*

In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    &   ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Theorem

usgriedgdomord 16017*

Alternate version of usgredgdomord 16022, not using the notation (Edg‘𝐺). In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ≼ 𝑉)

Theorem

ushgredgedg 16018*

In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)

⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Theorem

usgredgedg 16019*

In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)

⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Theorem

ushgredgedgloop 16020*

In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex 𝑁 and the set of loops at this vertex 𝑁. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.)

⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Theorem

uspgredgdomord 16021*

In a simple pseudograph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ≼ 𝑉)

Theorem

usgredgdomord 16022*

In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ≼ 𝑉)

Theorem

usgrstrrepeen 16023*

Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)

⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐼 = (.ef‘ndx)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})    ⇒   ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)

12.3  Walks, paths and cycles

12.3.1  Walks

Syntax

cwlks 16024

Extend class notation with walks (i.e. 1-walks) (of a hypergraph).

class Walks

Definition

df-wlks 16025*

Define the set of all walks (in a hypergraph). Such walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" discussed in Aksoy et al. The predicate 𝐹(Walks‘𝐺)𝑃 can be read as "The pair ⟨𝐹, 𝑃⟩ represents a walk in a graph 𝐺", see also iswlk 16029.

The condition {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝‘𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝‘𝑘).

According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by AV, 30-Dec-2020.)

⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})

Theorem

wkslem1 16026

Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)

⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵)))))

Theorem

wkslem2 16027

Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)

⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))

Theorem

wksfval 16028*

The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))})

Theorem

iswlk 16029*

Properties of a pair of functions to be/represent a walk. (Contributed by AV, 30-Dec-2020.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Theorem

wlkpropg 16030*

Properties of a walk. (Contributed by AV, 5-Nov-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))

Theorem

wlkvg 16031

The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.)

⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → (𝐹 ∈ V ∧ 𝑃 ∈ V))

Theorem

iswlkg 16032*

Generalization of iswlk 16029: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Theorem

wlkfg 16033

The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹 ∈ Word dom 𝐼)

Theorem

wlkclg 16034

A walk has length ♯(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → (♯‘𝐹) ∈ ℕ₀)

Theorem

wlkpg 16035

The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → 𝑃:(0...(♯‘𝐹))⟶𝑉)

Theorem

wlkpwrdg 16036

The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → 𝑃 ∈ Word 𝑉)

Theorem

wlklenvp1g 16037

The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)

⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → (♯‘𝑃) = ((♯‘𝐹) + 1))

Theorem

wlkm 16038*

The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.)

⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → ∃𝑥 𝑥 ∈ 𝑃)

Theorem

wlklenvm1g 16039

The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)

⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → (♯‘𝐹) = ((♯‘𝑃) − 1))

Theorem

ifpsnprss 16040

Lemma for wlkvtxeledgg 16041: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)

⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Theorem

wlkvtxeledgg 16041*

Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))

Theorem

wlkvtxiedgg 16042*

The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)

PART 13  GUIDES AND MISCELLANEA

13.1  Guides (conventions, explanations, and examples)

13.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 16043

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 16133 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5999, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5658 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Expression	Syntax?	Example(s)
ap	apart	df-pap 7430, df-ap 8725		Yes	apadd1 8751, apne 8766
g	with "is a set" condition			No	1stvalg 6286, brtposg 6398, setsmsbasg 15147
m	inhabited (from "member")		∃𝑥𝑥 ∈ 𝐴	No	r19.2m 3578, negm 9806, ctm 7272, basmex 13087
seq3, sum3	recursive sequence	df-seqfrec 10665		Yes	seq3-1 10679, fsum3 11893
tap	tight apartness	df-tap 7432		Yes	df-tap 7432

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7432.

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

13.1.2  Definitional examples

Theorem

ex-or 16044

Example for ax-io 714. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 3 ∨ 4 = 4)

Theorem

ex-an 16045

Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 16046

Example for df-dec 9575, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9575 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000

Theorem

ex-fl 16047

Example for df-fl 10485. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 16048

Example for df-ceil 10486. (Contributed by AV, 4-Sep-2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-exp 16049

Example for df-exp 10756. (Contributed by AV, 4-Sep-2021.)

⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9))

Theorem

ex-fac 16050

Example for df-fac 10943. (Contributed by AV, 4-Sep-2021.)

⊢ (!‘5) = ;;120

Theorem

ex-bc 16051

Example for df-bc 10965. (Contributed by AV, 4-Sep-2021.)

⊢ (5C3) = ;10

Theorem

ex-dvds 16052

Example for df-dvds 12294: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

⊢ 3 ∥ 6

Theorem

ex-gcd 16053

Example for df-gcd 12470. (Contributed by AV, 5-Sep-2021.)

⊢ (-6 gcd 9) = 3

PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1  Mathboxes for user contributions

14.1.1  Mathbox guidelines

Theorem

mathbox 16054

(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

14.2  Mathbox for BJ

14.2.1  Propositional calculus

Theorem

bj-nnsn 16055

As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

Theorem

bj-nnor 16056

Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)

⊢ (¬ ¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnim 16057

The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnan 16058

The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

Theorem

bj-nnclavius 16059

Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)

⊢ ((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)

Theorem

bj-imnimnn 16060

If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16059 as its last step. (Contributed by BJ, 27-Oct-2024.)

⊢ (𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ ¬ ¬ 𝜓

14.2.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16062 or bj-stfal 16064 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

Theorem

bj-trst 16061

A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (𝜑 → STAB 𝜑)

Theorem

bj-sttru 16062

The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)

⊢ STAB ⊤

Theorem

bj-fast 16063

A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ 𝜑 → STAB 𝜑)

Theorem

bj-stfal 16064

The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)

⊢ STAB ⊥

Theorem

bj-nnst 16065

Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 16311 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

⊢ ¬ ¬ STAB 𝜑

Theorem

bj-nnbist 16066

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16079). (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))

Theorem

bj-stst 16067

Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (STAB STAB 𝜑 ↔ STAB 𝜑)

Theorem

bj-stim 16068

A conjunction with a stable consequent is stable. See stabnot 838 for negation , bj-stan 16069 for conjunction , and bj-stal 16071 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓))

Theorem

bj-stan 16069

The conjunction of two stable formulas is stable. See bj-stim 16068 for implication, stabnot 838 for negation, and bj-stal 16071 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ ((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))

Theorem

bj-stand 16070

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16069. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16069 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

⊢ (𝜑 → STAB 𝜓)    &   ⊢ (𝜑 → STAB 𝜒)    ⇒   ⊢ (𝜑 → STAB (𝜓 ∧ 𝜒))

Theorem

bj-stal 16071

The universal quantification of a stable formula is stable. See bj-stim 16068 for implication, stabnot 838 for negation, and bj-stan 16069 for conjunction. (Contributed by BJ, 24-Nov-2023.)

⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)

Theorem

bj-pm2.18st 16072

Clavius law for stable formulas. See pm2.18dc 860. (Contributed by BJ, 4-Dec-2023.)

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Theorem

bj-con1st 16073

Contraposition when the antecedent is a negated stable proposition. See con1dc 861. (Contributed by BJ, 11-Nov-2024.)

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

14.2.1.2  Decidable formulas

Theorem

bj-trdc 16074

A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

⊢ (𝜑 → DECID 𝜑)

Theorem

bj-dctru 16075

The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

⊢ DECID ⊤

Theorem

bj-fadc 16076

A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ 𝜑 → DECID 𝜑)

Theorem

bj-dcfal 16077

The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

⊢ DECID ⊥

Theorem

bj-dcstab 16078

A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

⊢ (DECID 𝜑 → STAB 𝜑)

Theorem

bj-nnbidc 16079

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16066. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Theorem

bj-nndcALT 16080

Alternate proof of nndc 856. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ DECID 𝜑

Theorem

bj-dcdc 16081

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

Theorem

bj-stdc 16082

Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)

⊢ (STAB DECID 𝜑 ↔ DECID 𝜑)

Theorem

bj-dcst 16083

Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

14.2.2  Predicate calculus

Theorem

bj-ex 16084*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1644 and 19.9ht 1687 or 19.23ht 1543). (Proof modification is discouraged.)

⊢ (∃𝑥𝜑 → 𝜑)

Theorem

bj-hbalt 16085

Closed form of hbal 1523 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

bj-nfalt 16086

Closed form of nfal 1622 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)

⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Theorem

spimd 16087

Deduction form of spim 1784. (Contributed by BJ, 17-Oct-2019.)

⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Theorem

2spim 16088*

Double substitution, as in spim 1784. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑧𝜒    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Theorem

ch2var 16089*

Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

ch2varv 16090*

Version of ch2var 16089 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

bj-exlimmp 16091

Lemma for bj-vtoclgf 16098. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Theorem

bj-exlimmpi 16092

Lemma for bj-vtoclgf 16098. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)

Theorem

bj-sbimedh 16093

A strengthening of sbiedh 1833 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))

Theorem

bj-sbimeh 16094

A strengthening of sbieh 1836 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Theorem

bj-sbime 16095

A strengthening of sbie 1837 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

14.2.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 16096

Shorter proof of el2oss1o 6587 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 2_o → 𝐴 ⊆ 1_o)

14.2.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 16097

Weakening two hypotheses of vtoclgf 2859. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Theorem

bj-vtoclgf 16098

Weakening two hypotheses of vtoclgf 2859. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Theorem

elabgf0 16099

Lemma for elabgf 2945. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))

Theorem

elabgft1 16100

One implication of elabgf 2945, in closed form. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))